Algebra and Trigonometry 2e2-38

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rectangular grid (see Figure 12). Just as is the standard form for a vertical line in rectangular form, is
the standard form for a vertical line in polar form.
Figure 12 (a) Polar grid (b) Rectangular coordinate system
A similar discussion would demonstrate that the graph of the function will be the horizontal line In
fact, is the standard form for a horizontal line in polar form, corresponding to the rectangular form
EXAMPLE 10
Rewriting a Polar Equation in Cartesian Form
Rewrite the polar equation as a Cartesian equation.
Solution
The goal is to eliminate and and introduce and We clear the fraction, and then use substitution. In order to
replace with and we must use the expression
The Cartesian equation is However, to graph it, especially using a graphing calculator or computer
program, we want to isolate
When our entire equation has been changed from and to and we can stop, unless asked to solve for or simplify.
See Figure 13.
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Figure 13
The “hour-glass” shape of the graph is called a hyperbola. Hyperbolas have many interesting geometric features and
applications, which we will investigate further in Analytic Geometry.
Analysis
In this example, the right side of the equation can be expanded and the equation simplified further, as shown above.
However, the equation cannot be written as a single function in Cartesian form. We may wish to write the rectangular
equation in the hyperbola’s standard form. To do this, we can start with the initial equation.
TRY IT #5 Rewrite the polar equation in Cartesian form.
EXAMPLE 11
Rewriting a Polar Equation in Cartesian Form
Rewrite the polar equation in Cartesian form.
Solution
This equation can also be written as
10.3 • Polar Coordinates 935
MEDIA
Access these online resources for additional instruction and practice with polar coordinates.
Introduction to Polar Coordinates (http://openstax.org/l/intropolar)
Comparing Polar and Rectangular Coordinates (http://openstax.org/l/polarrect)
10.3 SECTION EXERCISES
Verbal
1. How are polar coordinates
different from rectangular
coordinates?
2. How are the polar axes
different from the x- and
y-axes of the Cartesian
plane?
3. Explain how polar
coordinates are graphed.
4. How are the points
and related?
5. Explain why the points
and are
the same.
Algebraic
For the following exercises, convert the given polar coordinates to Cartesian coordinates. Remember to consider the
quadrant in which the given point is located when determining for the point.
6. 7. 8.
9. 10.
For the following exercises, convert the given Cartesian coordinates to polar coordinates with
Remember to consider the quadrant in which the given point is located.
11. 12. 13.
14. 15.
For the following exercises, convert the given Cartesian equation to a polar equation.
16. 17. 18.
19. 20. 21.
22. 23. 24.
25. 26. 27.
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For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a
conic if possible, and identify the conic section represented.
28. 29. 30.
31. 32. 33.
34. 35. 36.
37. 38. 39.
Graphical
For the following exercises, find the polar coordinates of the point.
40. 41. 42.
43. 44.
For the following exercises, plot the points.
45. 46. 47.
48. 49. 50.
10.3 • Polar Coordinates 937
51. 52. 53.
54.
For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.
55. 56. 57.
58. 59. 60.
61.
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.
62. 63. 64.
65. 66. 67.
68.
Technology
69. Use a graphing calculator
to find the rectangular
coordinates of
Round to the nearest
thousandth.
70. Use a graphing calculator
to find the rectangular
coordinates of
Round to the nearest
thousandth.
71. Use a graphing calculator
to find the polar
coordinates of in
degrees. Round to the
nearest thousandth.
72. Use a graphing calculator
to find the polar
coordinates of in
degrees. Round to the
nearest hundredth.
73. Use a graphing calculator
to find the polar
coordinates of in
radians. Round to the
nearest hundredth.
Extensions
74. Describe the graph of 75. Describe the graph of 76. Describe the graph of
77. Describe the graph of 78. What polar equations will
give an oblique line?
For the following exercise, graph the polar inequality.
79. 80. 81.
938 10 • Further Applications of Trigonometry
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